Work of Song Dai
joint with Tianqi Wu
Let S be a surface with a triangulation T . A PL metric on T is a positive-valued function on the edge set such that each face forms a Euclidean triangle. The discrete curvature at a vertex is defined as 2π minus the sum of the angles around that vertex. Analogous to the smooth case, Luo introduced the concept of discrete conformality, which involves considering a positive-valued function on the vertex set and multiplying the length of each edge by the product of the function values at its two endpoints. Bobenko-Pinkall-Springborn discovered that by endowing each face with a hyperbolic metric under the Klein model, the surface becomes a hyperbolic surface with cusps. For two PL metrics, they are discretely conformal if and only if the induced hyperbolic surfaces are isometric.
A fundamental question is whether, for a fixed triangulation, a PL metric exhibits rigidity with respect to discrete curvature, that is, whether two PL metrics yielding the same curvature differ only by a scaling factor. We show that if the considered PL metric is an isometric subdivision of the Euclidean plane and satisfies the Delaunay condition and nondegenerate condition, then the rigidity holds. This result corresponds to the Cauchy rigidity for certain non-compact ideal hyperbolic polyhedra. Roughly speaking, the geometry of the boundary of a convex body decides the geometry of the whole body.
Advances in Mathematics
https://doi.org/10.1016/j.aim.2024.109910
